广义凸函数的上图与水平集

Epigraph and Level Set of Generalized Convex Functions

【目的】对于广义凸集与广义凸函数的研究是十分基础而重要的课题,只有广义凸性的基础理论研究不断完善,对广义凸规划的全面深入的研究才会成为可能。【方法】受凸函数的一个基本结果启发,给出概念并借助相关结果开展论证。【结果】简化了已有文献所给出的广义凸函数概念,指出函数的广义凸性与函数上图的广义凸性之间的等价关系,并给出下半连续前提下,F-G广义凸函数与F-G广义弱凸函数之间等价性的新证明。最后,指出函数的广义凸性与函数水平集的广义凸性之间的内在联系。【结论】将广义凸集和广义凸函数统一在一个结构框架下进行研究,建立了二者之间的桥梁纽带。

[Purposes]The study of generalized convex sets and generalized convex functions are very important,because the developments of generalized convex theory can provide the possible to study the generalized convex programming.[Methods]Some concepts and proofs were given by the results of convex functions.[Findings]First,the concepts of generalized convex functions were simplified and the equivalent relationships between generalized convexity of functions and epigraph were introduced.Second,base on the condition of lower semi-continuous,the new proof of equivalent relationships between F-Ggeneralized convex and F-G generalized weakly convex functions was given.Last,the relationships between generalized convexity of functions and level sets were introduced.[Conclusions]Generalized convex sets and generalized convex functions were studied in a common structural framework,and a link was established between them.